Space l_infty(R) with sup norm

[bugatti79]

Homework Statement

Prove that the space l_\infty (R) of bounded sequences with the sup norm (ie x=(x_n) in l_\infty (R) )is not a inner product space.

The Attempt at a Solution

Using definition of parallelogram ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) (1)

Consider x_n=1^-n and y_n=4-1/n then

||x||_\infty=sup|x_n| = 1 where n=1 and |y||_\infty=sup|y_n|=4 where n=\infty

||x+y||_\infty=|1+4|=5 where I chose n=1 for x_n and n=\infty for y_n so that we get the maximum value.

||x-y||_\infty = |0-4|=4 where I chose n=\infty for x_n and n=1 for y_n to get a maximum

||x+y||^2_\infty+||x-y||^2_\infty=41

THis is not equal to the RHS 2(||x||^2+||y||^2)= 34

implies the space l_\infty(R) is not an inner product space..?

 

[mighty2000]

Thank you for your post. I would like to provide a more detailed explanation to show why l_\infty(R) is not an inner product space.

Firstly, an inner product space is a vector space equipped with an inner product, which is a function that takes in two vectors and returns a scalar value. This inner product must satisfy certain properties, such as linearity, symmetry, and positive definiteness. In the case of l_\infty(R), the sup norm ||x||_\infty does not satisfy the linearity property, as shown in your proof. This means that it cannot be considered as an inner product.

Furthermore, in an inner product space, the inner product can be used to define a norm, which is a function that maps a vector to a non-negative scalar value. The norm must also satisfy certain properties, such as non-negativity, homogeneity, and the triangle inequality. However, in l_\infty(R), the sup norm does not satisfy the triangle inequality, as shown in your proof. This means that it cannot be used to define a norm, which is a crucial property of an inner product space.

In conclusion, l_\infty(R) cannot be considered as an inner product space due to the violation of fundamental properties of an inner product and a norm. Thank you for bringing up this interesting topic for discussion.